Extending property on EC-Fully Submodules

Abstract

There are several generalizations of -modules in literature. One of the generalization is based on fully invariant submodules. Recall that a module is called -extending if every fully invariant submodule is essential in a direct summand. We call a module -extending if every fully invariant submodule which contains essentially a cyclic submodule is essential in a direct summand. Initially we obtain basic properties in the general module setting. For example, a direct sum of -extending modules is -extending. Again, like the -extending property, the -extending property is shown to carry over to matrix rings.